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Theorem oteq2 4069
Description: Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.)
Assertion
Ref Expression
oteq2  |-  ( A  =  B  ->  <. C ,  A ,  D >.  = 
<. C ,  B ,  D >. )

Proof of Theorem oteq2
StepHypRef Expression
1 opeq2 4060 . . 3  |-  ( A  =  B  ->  <. C ,  A >.  =  <. C ,  B >. )
21opeq1d 4065 . 2  |-  ( A  =  B  ->  <. <. C ,  A >. ,  D >.  = 
<. <. C ,  B >. ,  D >. )
3 df-ot 3886 . 2  |-  <. C ,  A ,  D >.  = 
<. <. C ,  A >. ,  D >.
4 df-ot 3886 . 2  |-  <. C ,  B ,  D >.  = 
<. <. C ,  B >. ,  D >.
52, 3, 43eqtr4g 2500 1  |-  ( A  =  B  ->  <. C ,  A ,  D >.  = 
<. C ,  B ,  D >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1369   <.cop 3883   <.cotp 3885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-rab 2724  df-v 2974  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-ot 3886
This theorem is referenced by:  oteq2d  4072  efgi  16216  efgtf  16219  efgtval  16220  el2wlkonot  30388  el2spthonot  30389  frg2wot1  30650  usg2spot2nb  30658  mapdh9aOLDN  35436  hdmap1eulemOLDN  35470
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